The simple answer is no. The energy needed to push extra two electrons next to a hydrogen atom are enough to make all three electrons fly away. But you can sort of do it. It's called Hydrogen Bonding
A positive Hydrogen Ion is formed by losing its electron. Adding an electron makes a negative Hydrogen Ion which is only found in compounds with highly electropositive metals such as Sodium and Potassium. (Sodium Hydride and Potassium Hydride)
Rather than making a negative ion Hydrogen normally forms covalent bonds by sharing electrons.
In water each Hydrogen shares a pair of electron with the Oxygen atom. (One electron from each atom) So it fills its inner shell without gaining a charge - that is it has half ownership of two electrons. The Oxygen uses the shared electrons to fill its outer shell. That is it has 4 electrons of its own and 4 that it is sharing with the two hydrogen.
However a hydrogen atom in water has its electron (or more properly its two half electrons) pulled to one side by the oxygen atom. This exposes the positive nucleus. Normally nuclei are shielded by an electron screen. But hydrogen in water, although it is electrically neutral acts a bit like it is positively charged. Consequently it can pull on electrons around the oxygen atom of a nearby water molecule. This is called a hydrogen bond.
Hydrogen bonds are responsible for water's weird behavior. Ice floats and the boiling point is way too high. Compare H2S which boils at -60 °C.
If you look at a hydrogen atom in water, it has half shares in two covalent electrons and a stake in a third forming the hydrogen bond. So it does actually have three electrons. Not the answer you wanted but it sort of works.